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Title

β -delayed neutron-emission and fission calculations within

relativistic quasiparticle random-phase approximation and astatistical model

Author(s)Minato Futoshi, Marketin T., Paar N.

Citation

Physical Review C, 104(4), p. 044321_1‐044321_18

Text Version

Published Journal Article

URL

https://jopss.jaea.go.jp/search/servlet/search?5070228

DOI

https://doi.org/10.1103/PhysRevC.104.044321

Right©2021 American Physical Society

Japan Atomic Energy Agency

PHYSICAL REVIEW C 104, 044321 (2021)

β-delayed neutron-emission and fission calculations within relativistic quasiparticle
random-phase approximation and a statistical model
Futoshi Minato ,1,* Tomislav Marketin ,2 and Nils Paar2
1

2

Nuclear Data Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia(Received 23 February 2021; revised 21 July 2021; accepted 23 September 2021; published 19 October 2021)
Background: β-delayed neutron emission and fission are essential in r-process nucleosynthesis. Although thenumber of experimental studies covering r-process nuclei has recently increased, the uncertainties of β-delayed
neutron emission and fission are still large for r-process simulations.
Purpose: Our aim is to introduce a theoretical framework for the description of β-delayed neutron-emission andfission rates based on relativistic nuclear energy density-functional and statistical models and investigate theirproperties throughout the nuclide map.
Methods: To obtain β strength functions, the relativistic proton-neutron quasiparticle random-phase approximation is employed. Particle evaporations and fission from highly excited nuclear states are estimated by the
Hauser-Feshbach statistical model. β-delayed neutron branching ratios Pn are calculated and compared withexperimental data, and the β-delayed fission branching ratio Pf are also assessed by using different fission
barrier data.
Results: Calculated Pn are in a good agreement with the experimental data and the root mean square deviationis comparable to results of preceding works. It is found that energy withdrawal by β-delayed neutron-emission
sensitivity varies Pn , especially for nuclei near the neutron drip line. Pf depend sensitively on fission barrier data.
It is found that not only the barrier height but also the number of barrier humps is important to evaluate Pf .
Conclusions: The framework introduced in this work provides an improved theoretical description of the
β-delayed neutron emission and fission. Since Pf as well as Pn depend strongly on fission barrier information,
four kinds of fission barrier data are used in this work to allow further sensitivity studies of the r-process
nucleosynthesis on the nuclear fission. More studies on fission barrier are highly requested to assess the roleof β-delayed fission in the r-process study. A complete set of calculated data for β-delayed neutron emission
and fission are summarized as a table in supplemental material for its use in r-process studies as well as tocomplement a part of nuclear data in which no experimental data are available.
DOI: 10.1103/PhysRevC.104.044321

I. INTRODUCTION

The origin of chemical elements in the universe is the longstanding problem and one of the hottest topics in astrophysics.
Our knowledge accumulated so far indicates that heavy elements in nature are generated by the dynamical processesof stars. In particular, about a half of the elements heavier
than iron is considered to be produced by the r process [1,2](the other half is by the s process [1]). Although it is still notconcluded where the r process occurs, recent studies employing observation of gravitational waves provide evidence that aneutron-star merger is one of the possible sites of the r process[3,4].
The solar r-process abundance pattern shows a characteristic mass distribution that has three peaks around A = 80–90,
130–138, and 195–208 [5–7]. The r-process simulation suggests that the origin of these peaks is related to neutron magic
numbers of N = 50, 82, and 126, which is also related to*minato.futoshi@jaea.go.jp

2469-9985/2021/104(4)/044321(18)nuclear mass [5,6,8]. However, the peak positions and abundance ratios cannot be reproduced only by considering thenuclear mass effects. Neutron capture, β decay, and otherdecay modes sensitively influence the abundance pattern of
the r process [6,9–27].
During the r process, β − decay (hereinafter, we simply call it β decay) increases the atomic number of nuclei
and produces daughter nuclei in a highly excited state. Depending on the excitation energy, the daughter nuclei decay
through several particle-emission channels. In particular, βdelayed neutron emission and fission (hereinafter we callthem BDNE and BDF, respectively) play a subsidiary rolein the r-process abundance. BDNE produces two different
effects on the r-process abundance. The first is that it leads
nuclei in an r-process site to detour the β − -decay path
(A, Z ) → (A, Z + 1) by reducing their neutron number, forexample, (A, Z ) → (A − 1, Z + 1) in case of one-neutron(1n) emission, where Z equals the proton number. This effect smooths the even-odd fluctuation in the final r-process
abundance pattern [19–21]. Another effect of BDNE is thatit feeds neutrons to the r-process environment. This effect044321-1©2021 American Physical Society

MINATO, MARKETIN, AND PAAR

PHYSICAL REVIEW C 104, 044321 (2021)

retards the progress of the freeze-out for the “cold” r-process
scenario [6].
The role of BDF is to reduce the abundance of heavy elements by breaking daughter nuclei into two or more fragments
during the r process and its freeze-out phase. Furthermore,
the fission fragments restart to capture environmental neutrons
and grow up toward heavy elements again. This phenomena,
also known as fission recycling, affects a wide range of the
r-process abundances together with neutron-induced fission[22–27]. An understanding of fission in the r-process is essential to answer the naive question, are superheavy elements
produced by dynamical processes in stars [28]?
A lot of experimental measurements of BDNE have been
carried out because of its importance in applications to ther-process as well as for nuclear data evaluation (e.g., see
Refs. [29–33] for recent works). However, the experimental
difficulty rapidly increases as one tries to study very-neutronrich nuclei because of the low statistics. For this reason, there
are still nuclei for which BDNE and BDF have not beenmeasured yet. In particular, for BDF, only seven cases arerecognized near the β-stability line [34] (see Ref. [35], whichsummarizes experimental studies for BDF).
Nuclei that are not investigated by experiment have to
be covered by theoretical models. Empirical systematics isa useful approach and is provided in, e.g., Refs. [36–39]
for the β-delayed neutron branching ratio Pn and even for
BDF in Refs. [35,40]. Gross Theory (GT2) [41,42] is also aneffective tool. It has provided β strength functions systematically for nuclei in the nuclear chart, and not only half-lives
but also Pn and the β-delayed fission branching ratio Pf
have been calculated [20]. However, it is not clear whether
a phenomenological approach based on the GT2 is valid forvery-neutron-rich nuclei.
Another effective approach for theoretically predicting
Pn and Pf is a microscopic model. Typical approaches are
the configuration interaction (CI) model, the quasiparticle
random-phase approximation (QRPA), and the finite amplitude method (FAM), which is essentially the same as the
QRPA but solves the problem in a different way. Recently,
the interacting boson model (IBM) based on the mean-fieldapproach has been applied to β-decay calculations as well[43]. Although CI models are actively applied to the calculation of half-lives for r-process nuclei around N = 50, 82, and126 shell closures, the application is still restricted to limited
nuclei due to the increasing computational cost [14,44,45].
For this reason, to calculate β decay for r-process nuclei, the
QRPA [46–51] and FAM [52–54] are applied in practice.
One of us has calculated Pn systematically for neutronrich nuclei in the framework of the proton-neutron relativistic
quasiparticle random-phase approximation (pn-RQRPA) [15].
This microscopic model starts from an effective interaction
determined by experimentally known ground-state propertiesof nuclei and is believed to provide a more reliable β strength
function than purely phenomenological approaches. To predict β-delayed neutron branching ratios, a simplified approach(hereinafter referred to as the cutoff method) assuming that
nuclei with excitation energies above the neutron thresholdalways emit β-delayed neutron has been used, as done in
Refs. [48–50]. This assumption corresponds with a picturethat neutron emission is the only exit channel above neutron
thresholds and the kinetic energy of β-delayed neutrons iszero. This prescription clearly omits the nuclear structure,
the selection rule of the decay chain, competition with otherdecay channels, and kinematics. In fact, it is pointed out in
Ref. [55] that a competition between neutron emission and γ
deexcitation gives a non-negligible variation to the calculation
for Pn of nuclei, especially near the neutron drip line.
One of the approaches to treat nuclear decay in a more
physical and complete way is to apply a statistical decay
model, for example, the Hauser-Feshbach statistical model(HFM) [56]. The HFM considers nuclear structure effects
through level densities and selection rules of the decay chain,competition with other decay channels, and kinematics thatthe cutoff method omitted. Combination of the QRPA β
strength function and the statistical decay model is therefore a feasible approach and has been carried out by several
groups using FRDM + QRPA [26,57,58] and nonrelativistic
QRPA [46,47,51,59–61]. The aim of this paper is directed
at estimating Pn and Pf by using this approach, namely, by
combining pn-RQRPA and HFM. Hereinafter, we refer to thepresent work as pn-RQRPA + HFM to distinguish it from thepreceding work of pn-RQRPA [15].
This paper is organized as follows: In Sec. II, we describe
theoretical framework to calculate BDNE and BDF using
pn-RQRPA + HFM. In Sec. III, the results obtained in thiswork are presented and discussed in comparison with experimental data and preceding works. Section IV summarizes thiswork and presents some perspectives. The complete data tablecontaining the BDNE and BDF branching ratios is available
in the Supplemental Material [62].

II. THEORETICAL FRAMEWORK OF pn-RQRPA + HFM
A. β-delayed neutron and fission branching ratios

Our calculation is composed of two parts: First, we prepare
β strength functions for the Gamow-Teller (GT) and the firstforbidden (FF) transitions by using the pn-RQRPA [15]. As
second step, we carry out the calculation of statistical decay
from the compound state by using the HFM calculation withexcitation energy and spin-parity given by the pn-RQRPA. Pn
and Pf are then obtained by multiplying neutron and fission
emission probabilities by the β-decay rates, respectively.
A fully self-consistent covariant density-functional theory(CDFT) is adopted in this work. The ground state of all nuclei
is calculated with the relativistic Hartree-Bogoliubov (RHB)
model with the D3C* interaction [63]. The ground state of odd
nuclei are computed by employing the same model as that ofeven-even nuclei, namely, we impose the expectation valueof the particle number operator to be an odd proton and/or
neutron number. On the top of the RHB model, excited states
are obtained within the pn-RQRPA. More details about thepn-RQRPA used in this work are given in Ref. [15].
We assume that daughter nuclei reach the compound state,
namely, the thermally equilibrium state, soon after the β decay. In the compound state, daughter nuclei lose their initial
information that they had before the β decay, except for the
spin-parity and the total energy of the system. The number

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PHYSICAL REVIEW C 104, 044321 (2021)

of protons and neutrons of the initial nuclei (precursor) aredefined as Z and N, respectively. Accordingly, the number ofprotons and neutrons of the daughter nuclei are given by Z + 1
and N − 1, respectively.
The HFM calculation is executed with various different
excitation energies and spin-parity states of daughter nuclei.
However, excitation energies Ei,QRPA computed from the pnRQRPA are not identical to those of daughter nuclei. We thus
need some transformations. In the framework of pn-RQRPA,
coherent proton-neutron two-quasiparticle configurations are
regarded as excited states, where the lowest state becomes theground state of the daughter nuclei. Namely, the excitationenergies of the pn-RQRPA has to be subtracted from the lowest value to obtain those with respect to the daughter nuclei.
However, it is time consuming and not straightforward tofind the lowest state within the pn-RQRPA. As an alternative
method to connect the pn-RQRPA and the HFM calculations,
we use in this work the noninteracting quasiparticle approximation, in which an odd-mass nucleus is approximated by one
quasiparticle state plus the wave function of the neighboringeven-mass nucleus. Based on this approximation, the excitation energies with respect to daughter nuclei are computed
through [64,65]
Ei∗ = Ei,QRPA − Ecorr ,⎧E + En0⎪⎨ p0
E p0
Ecorr =
⎪En0⎩0

(β decay for even-even nucleus)(for even-odd)(for odd-even)(for odd-odd),

(1)

(2)

where the index i denotes an excited state of the daughter
nuclei with spin-parity J π , and E p0 and En0 are the lowest
quasiparticle energies of proton and neutron calculated fromthe RHB, respectively.
We should mention one issue arising from the use of thenoninteracting quasiparticle approximation in Eq. (1). The
pn-QRPA excitation energies Ei,QRPA consider correlations
caused by the two-body residual interactions, while the lowest
quasiparticle energies in Eq. (2) does not. As a result, Ei,QRPA
is frequently lower than Ecorr , and Ei∗ becomes negative forsome nuclei. Although negative excitation energies are physically incorrect, we adopt an approximation that the HFM
calculation with Ei∗ < 0 MeV ends up with no evaporationfrom nuclei and the corresponding states become the groundstate.
Carrying out the HFM calculation, we obtain production
ratios of evaporation residues with proton number Z  = Z + 1∗
and neutron number N  = N − 1 − x defined as p(n)xn (Ei ),and spectra of emitted particles defined as dν (Eν , Ei∗ ). Here,Eν is the kinetic energy of the outgoing particle and ν ={n, γ , p, α} represents a kind of the emitted particle, where theletters in the brackets represent neutron, γ ray, proton, and α
particle, respectively. We did not consider other light-particle
emissions d, t, and 3 He because they are strongly hindered
for the neutron-rich nuclei of interest. For heavy nuclei, thefission channel, i.e., BDF, is open. BDF occurs directly afterβ − decay of parent nuclei, i.e., (β − , f ), or indirectly after
multineutron emissions following the β decay, i.e., (β − , xn f ).
The fission probabilities from an excited state i are definedas p(xnf ) (Ei∗ ). The case of x = 0 means fission occurs directlyfrom the daughter nucleus with no neutron emission. The
(f )
∗
∗
∗
functions of p(n)xn (Ei ), pxn (Ei ), and dν (Eν , Ei ) are computedby the HFM calculation implemented in the CCONE code [66].
Although we do not go into details about the HFM because
the formalism is given in, e.g., Refs. [55,60,66], we will explain the nuclear input details used in our calculation in theSec. II B.
The BDNE and BDF branching ratios are calculated by

1
∗
Pn =
(3)
Pxn =
ri p(n)xn (Ei )
R
x
i,x
and
Pf =



1
ri p(xnf ) (Ei∗ ),R i,xPxn, f =x(4)

respectively, where ri is the partial β − -decay rates to excited state i calculated by the pn-RQRPA, and R = i ri .
β-delayed α (p) emission branching ratios Pα and Pxnα (Ppand Pxnp ) are calculated in the same way. In this framework,
the following relations hold:
Pn + Pf + Pα + Pp = 1.
The BDNE spectrum is calculated byri dn (En , Ei∗ ),Dn (En ) = Ns

(5)

(6)

i

where Ns is the normalization factor for spectra to be determined so as to satisfy ∞Dn (En ) dEn = 1.
(7)0

The summation of i of Eqs. (3), (4), and (6) is carried out
for Ei∗  Qβ . The β-decay Q value is calculated from Qβ =MnH + λn − λ p − Ecorr [15] where MnH , λn , and λ p are themass difference between neutron and hydrogen atom, and theneutron and proton Fermi energies, respectively.
B. Nuclear input details for Hauser-Feshbach statistical model
1. β strength function

The present RHB model is solved by using the D3C∗ interaction, resulting in discrete quasiparticle states obtained in
the canonical basis [63]. The pn-RQRPA equation is solved
by assuming coherent two quasiparticle (2qp) excitations. In
this scheme, the excited states are also discrete. Accordingly,
the β strength function also has a discrete shape in terms of
excitation energy. However, it is considered that actual excitedstates have a broad distribution with a width because of coupling to higher-order configurations (4qp, ..., nqp excitations)[67] and continuum states, which are not taken into account in
the present pn-RQRPA. To account for those influences, we
introduce a weight function to the β strength functions. We
consider two types of weight functions, that is, the Gaussian
type and the Lorentzian type:
⎧
∗ 2
i )
⎨  gw r √ 1 e− (E −E(8)2 2i∈w J i 2π 
Ww (E ) = 
/2
⎩
w 1
(9)
i∈w Ni gJ ri π (E −Ei∗ )2 +(/2)2 .

044321-3

MINATO, MARKETIN, AND PAAR

Strength (arb. unit)

10 0 1.0
0.8
0.6
0.4
0.2
10 -1 0.0

PHYSICAL REVIEW C 104, 044321 (2021)
Gaussian
Lorentzian8

9 10 11 12

2

4

10 -2

10 -3

10 -4

0

6

8
10 12
Energy (MeV)

14

16

18

20

FIG. 1. Weight functions of Gaussian type (the dotted line) of
Eq. (8) and Lorentzian (the solid line) type of Eq. (9) with E ∗ =10 MeV and width  = 0.6 MeV. The inserted panel is depicted in a
linear scale

The index w is used to distinguish the GT ( J π = 1+ ) andthe FF transitions ( J π = 0− , 1− , 2− ), and the factor gwJ
is a statistical factor that will be explained later. Since theLorentzian function that is proportional to 1/E 2 has a finite
strength even far from the mean, we introduce a cutoff energy
Ecut to reduce anomalously large strengths at distant energies.
This cutoff energy is determined by Lω (Ecut ) = Lω (Ei∗ )/1000,
and the weight function of Eq. (9) is active within the energy range of Ei∗ ± Ecut . The factor Ni in Eq. (9) is thenintroduced to renormalize the Lorentzian function to unity.
In this scheme, a part of the β strength functions may strayto the negative energies in terms of daughter nuclei. In thiscase, we integrate the β strengths at negative energies and set
them at E ∗ = 0 MeV, namely, the ground state of the daughter
nuclei. A schematic picture that depicts Eqs. (8) and (9) withr = 1, E ∗ = 10 MeV, and  = 0.6 MeV (Ecut = 9.5 MeV) are
shown in Fig. 1. Within E ∗ ± 2, the Gaussian function has abroader distribution than the Lorentzian function. Beyond thisenergy range, the Gaussian function rapidly fades out while
the Lorentzian function still has a finite strength distribution.
From Eqs. (8) and (9), Eqs. (3), (4), and (6) are rewritten as

1 Qβ
(10)
Pxn =
Ww (E )p(n)xn (E )dE ,
R 0

1 Qβ
(11)
Ww (E )p(xnf ) (E )dE ,Px f =R 0
 Qβ
Ww (E )dn (En , E )dE ,(12)
Dn (En ) = N
0respectively. In the later section, we determine the width parameter  that minimizes the root mean square deviation of Pn
from the experimental data.
2. Spin and parity

In case of even-even nuclei, the spin-parity of the groundstate is always 0+ and that of the daughter nuclei is uniquelydetermined according to the decay types. In case of odd mass
nuclei, the situation becomes complicated a little because thespin of the ground state is not always zero and the angularmomentum coupling is relevant to estimate the spin-parity ofdaughter nuclei. In this work, we use the following method
for odd-mass nuclei. First we determine the spin-parity of theground state of parent nuclei. We adopt the experimental data
if they are available. If not, we use the spin-parity of the statewith the lowest quasiparticle energy deduced from the RHB
calculation, namely, Jgs = j p with πgs = (−)l p and Jgs = jn
with πgs = (−)ln for odd-even and even-odd nuclei, respectively. Here, j p ( jn ) and l p (ln ) are the total and orbital angularmomentum of the proton (neutron), respectively. For the spinstate of odd-odd nuclei, we apply the Nordheim method [68],which is given by
Jgs = j p + jn

if

j p = l p ± 21

andjn = ln ± 21 ,Jgs = | j p − jn |if

j p = l p ± 21

andjn = ln ∓ 21 ,(13)

and the parity is computed via πgs = πgs(p) πgs(n) = (−)l p +ln . After determining the spin-parity of parent nuclei in this way,
we can conclude that of the daughter nuclei according tothe β-decay type. The spin of the daughter nuclei (J f ) thushas |Jgs − J|  J f  Jgs + J and the parity π f = πgs πα ,where J = 0, 1, 2 and πα = ± depending on the type ofβ transitions. We assumed the equal distribution of the βπstrength function for each J f f state. The factor gwJ in Eq. (9)is determined so that total amount of the β strength function
is conserved. For example, in the case of a 1± transition
from a parent nucleus with the J = 3/2 (1/2) state, J f =±5/2, 3/2, 1/2 (3/2, 1/2) and g13/2(1/2) = 1/3 (1/2).
In the HFM calculation, transmission coefficients of nucleons and α particles are calculated by the optical potentials
of Koning-Delaroche [69] and Avrigeanu [70], respectively.
For nuclear level densities, the Gilbert-Cameron method [71]with the Mengoni-Nakajima parameter [72] is adopted. Forγ strength functions, the enhanced generalized Lorentzianfunction [73] is used. Mass data are taken from the global
nuclear mass model [74] and used for calculating neutron
separation energies Sxn , Qβ , Qα and so on.
Transmission coefficients for fission are calculated as follows: We assume a double- or triple-humped parabolic barrier
with the barrier penetrability for each barrier calculated by the
Hill-Wheeler equation [75]. The transmission coefficients are
obtained by assuming that the fission process occurs throughthe transition states above the fission barrier. All transition
states were approximated by the level-density formula described above. The transmission coefficient of a single barrier
for the state having excitation energy E and spin-parity J π ,Ti (E , J π ) is calculated by ∞ρi ( , J π )
π
d ,
(14)
Ti (E , J ) =i−
1 + exp −2π E −V0h̄ωiwhere the subscripts i = A, B, and C indicate the inner, middle, and outer barriers, respectively, ρi ( , J π ) is
the level density at the saddle points, and Vi and h̄ωirepresent the height and curvature of the fission barrier,respectively. The transmission coefficients for nuclei with

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PHYSICAL REVIEW C 104, 044321 (2021)

III. RESULTS
A. β-delayed neutron emission

In the last section, the width parameter  is introduced tomake the β strength function a broad distribution. To determine the most likely , we estimate the root mean squared(rms) value of P1n , which is defined as
xn
=
σrms

Nexpt 
1 

Nexpt


log10

i

Pxn,i (c)Pxn,i (e)

2
,

(15)

(i)
(c)
where Nexpt is the number of experimental data and Pxn(i)and Pxn (e) are the β-delayed neutron branching ratios oftheoretical models and experiment of nucleus i, respectively.
Note that in Ref. [26] a linear scaled rms is adopted to discuss
the predictive power of P1n calculated by the FRDM + QRPA
+ HFM [58]. However, we adopted logarithmic scaled rms as
Eq. (15) because experimental Pxn extend from a small valueof 10−2 to 102 as like half-lives.
1n
Figure 2 shows σrms
as a function of width parameter  of
the Gaussian type of Eq. (8) (dashed line) and the Lorentzian
type of Eq. (9) (solid line). Experimental data are mainly
taken from ENSDF [34], however, we replace the data if a
new BDNE branching ratio reported in Ref. [82] is available.
Below  = 0.2 (0.4) MeV for the Lorentzian (Gaussian) type,1n
σrms
are more than 1.0, which means a factor of ten differencein P1n on average. This is because β feedings above one
neutron separation energy Sn are limited and P1n is mostly
1nunderestimated. Increasing , σrms
becomes smaller becausesome β strengths seep into energies above Sn and the majority of the calculated P1n comes close to the experimental
values. Although there exists a β strength function that escapes from the 1n emission energy window (Sn  E ∗ < S2n )to x-neutron (xn) ones (Sx  E ∗ < S(x+1)n ), this outflow is
generally smaller than the gain to the 1n emission energy
window because β-decay rates become lower with increasing
1nexcitation energies. Eventually, σrms
takes a minimum value
1nat  = 0.6 MeV for the Lorentzian type (σrms= 0.602) and at

1.2

Gaussian
Lorentzian

1.1
1
Vrms

double- and triple-humped barriers are approximated to be
T (E , J π ) = TA TC (TA + TC ) and T (E , J π ) = TA TB TC /(TA TB +TB TC + TC TA ), respectively.
Since predicted fission barrier data greatly vary amongfission barrier data as we will see, our calculation is carried out using four different fission barriers: HFB-14 [76],the Extended Thomas Fermi plus Strutinsky Integral (ETFSI)
method [77], the FRDM [78], and the Spherical Basis Method(SBM) [79]. The former two provide multihumped fission barrier data, while the latter two give only single-barrier information. Fission barrier heights of FRDM are relatively low, while
those of SBM are relatively high. The curvature parameter weused is h̄ωA ( h̄ωB ) = 1.04(0.60), 0.80(0.52), 0.65(0.45) MeV
for even-even, even-odd or odd-even, and odd-odd nuclei,
respectively, that are determined to reproduce fission crosssections of uranium isotopes [66]. In case of HFB-14, theinformation on barrier curvature, fission path, and the level
density at the saddle points is provided [80,81], so we usethem to calculate the fission transmission coefficients.

0.9
0.8
0.7
0.6
0.5
0

0.2

0.4

0.6

0.8

1

1.2

1.4

Width (MeV)
1n
FIG. 2. Estimated σrms
value given in Eq. (15) as a function of
the width  calculated with the Gaussian-type function (8) and theLorentzian type function (9).

1n
 = 1.2 MeV for the Gaussian type (σrms= 0.694). Above1nthe minimal points, σrms turn into a slow increase. This isbecause β strengths contribute too much at energies above Sn ,and a majority of calculated P1n becomes large as comparedwith experimental values.
The Lorentzian-type weight function gives a better re1nsult than the Gaussian one in terms of σrms
. It is difficult1nto explain this because the result of σrms
is complicatedly
convoluted by the many P1n data. However, it is consideredthat the little leakage of the β strength function extending
far outside of the mean of the Lorentzian function plays a significant role for the better agreement of P1n with experimental
data. In fact, the Gaussian-type weight function needs a wider
1nwidth than the Lorentzian type to give the minimal σrms
. In
the subsequent sections in this paper, we discuss BDNE and
BDF using the weight function of Lorentzian type with  =0.6 MeV, the width of which is reasonable as compared withthe experimental measurements of low-lying GT resonances
for stable tin isotopes investigated by (3 He, t) reactions [83].
1nTable I lists σrmsof the pn-RQRPA + HFM together withthe pn-RQRPA [15], Gross theory (GT2) [41,42], and FRDM
1n
+ QRPA + HFM [58]. We also list σrms
classified by four1n
was 0.798 fordifferent ranges of P1n . The total value of σrms
the pn-RQRPA. Note that, in the pn-RQRPA, the β strength
functions are weighted by a Lorentzian function using a widthof 130 keV that is determined so as to reproduce β-delayed
neutron yield of thermal-neutron-induced fission of 235 U. The
present result of pn-RQRPA + HFM greatly improves that of1n= 0.601, which is comparable to
pn-RQRPA, providing σrms
1nthat of the GT2 (σrms= 0.595) and the FRDM + QRPA +HFM (0.512). Especially, we obtained a remarkable improve1n
1nthroughout from 1  P1n  100. Although σrms
ment in σrms
is deteriorated in P1n < 1, the result of pn-RQRPA + HFM is
still slightly better than that of the FRDM + QRPA + HFM.
Figure 3(a) shows the ratio of P1n for the pn-RQRPA +HFM to that for the pn-RQRPA as a function of mass number
A. We can see that most of the data points are above unity, i.e.,most of P1n is increased by the present framework. The main
reason for the increments is the use of the large . Figures 3(b)044321-5

MINATO, MARKETIN, AND PAAR

PHYSICAL REVIEW C 104, 044321 (2021)

1n
TABLE I. Result of σrms
defined by Eq. (15) for different ranges of P1n (e) (%).

Model
pn-RQRPA + HFM
pn-RQRPAGT2
FRDM + QRPA + HFM
Number of nuclei

P1n (e) < 1

1  P1n (e) < 1010  P1n (e) < 50

50  P1n (e)  100

All P1n (e) data

0.952
0.857
0.852
1.084
50

0.446
0.727
0.656
0.442
92

0.570
0.925
0.464
0.482
91

0.317
0.460
0.320
0.281
34

0.601
0.798
0.595
0.512
267

and 3(c) show the ratio of calculated P1n to experimental
P1n (C/E) for the pn-RQRPA + HFM and the pn-RQRPA,
respectively. The underestimations of P1n (i.e., C/E < 1) forthe pn-RQRPA are improved for the pn-RQRPA + HFM. We
confirmed that 73% of P1n corresponding to 193 nuclei are
improved for the pn-RQRPA + HFM.
Figure 4 plots Pn in the N − Z plane, which are calculatedby the pn-RQRPA + HFM with HFB-14 fission-barrier data.
The magic numbers (Z, N = 20, 28, 50, 82, 126) and possible
magic number N = 184 are shown by the double lines. For
nuclei close to the valley of stability, Pn are almost zero. As theneutron number increases, Pn are enhanced due to small Sxn
and large Qβ of neutron-rich nuclei. We observe the odd-evendependence, especially for the Z direction in the bottom-rightsector from Z = 82 and N = 126 and the upper-left sector
from Z = 82 and N = 184. In general, looking at an isotonic
chain, Qβ of odd-Z nuclei are larger than those of neighboringeven-Z nuclei because of pairing correlations, while Sxn areless sensitive to Z numbers. As a consequence, we obtain
the result that Pn of odd-Z nuclei are larger than neighboringeven-Z nuclei, generating the odd-even dependence. As an
example, the set of [Qβ (MeV), Sn (MeV)] for 220 Ir (Z = 77)

Proton Number

C/E

C/E

Ratio

10 (a)Ratio of pn-RQRPA+HFM to pn-RQRPA
8
6
4
2
0
(b)pn-RQRPA+HFM
102
1
10
0
10
-1
10
10-2
-3
10
(c)pn-RQRPA
102
1
10
0
10
-1
10
10-2
-3
10
20
40
60
80
100 120 140 160
Mass Ais about [12.2, 2.3], while that for the neighboring nuclei
219
Os (Z = 76) and 221 Pt (Z = 78) are about [11.1, 1.7] and[9.9, 2.2], respectively. Around the mass regions 95 < Z <101 and 184 < N < 200, Pn displays a patchy pattern. In thisregion BDF plays a role competing with BDNE. We discuss it
later in Sec. III B.
One may notice that Pn abruptly decrease in a region fromZ ≈ 65, N ≈ 140 to Z ≈ 80, N ≈ 180 forming a “valley” inFig. 4. In this region the single-particle energy of neutron
1i11/2 state exceeds that of proton 1h11/2 states in terms of thecanonical single-particle basis, allowing the 0− FF transition
with a low excitation energy. This is also seen in the ratioof the FF β decay rates to the total decay rates (Fig. 12 ofRef. [15]). Since the β decays via the 0− transition mostly
feed levels lower than neutron threshold energies, a sudden
decrease of Pn occurs. However, with increasing neutron number, the energy difference of neutron 1i11/2 and proton 1h11/2
states extends, and 0− FF transitions involving this configuration feed to excited states higher than Sxn . Eventually, Pn takes
a high value again in the region close to the neutron drip line.
Figure 5 shows Pxn from x = 1 to 5 in the N-Z plane, which
are calculated with the HFB-14 fission barrier data. Since 1nemission occurs most easily for neutron-rich nuclei at excitedstates, P1n distribute in a wide range of the N-Z plane. However, in a region close to the neutron drip line, 1n emissions
are no longer the major decay mode. Instead, multineutron
emissions become significant due to large Qβ and low Sxn .
Because the 2n emission channel is open at higher energy,
P2n extend in a more neutron-rich side than P1n . Similarly, thedistributions of Pxn gradually shift to the neutron drip line withincreasing x.
We observe that Pxn in Fig. 5 do not spread over a simple
straight band from low to heavy nuclei in the N-Z plane,

100
90
80
70
60
50
40
30
20
10

1):(
):($2
$2+
$2
$2+
2+1)
1):
1
))::3 m
attr
atr
a
trix
ix
’Pn.heat’ u ($1+1($1+1):($2+1):3
matrix

100
80
60
40

Pn (%)

20
0

20

FIG. 3. (a) Ratio of P1n for pn-RQRPA + HFM to that for
pn-RQRPA. Panels (b) and (c) represent ratio of calculated to experimental data of P1n (C/E) for pn-RQRPA + HFM (this work) and
pn-RQRPA [15], respectively.

40

60

80 100 120 140 160 180 200 220
Neutron NumberFIG. 4. Total β-delayed neutron branching ratios (Pn = x Pxn )calculated by the pn-RQRPA + HFM with HFB-14 fission barrier
data [76]. The black filled squares stand for stable or long-lived
nuclei.

044321-6

PHYSICAL REVIEW C 104, 044321 (2021)
100

100
90
80
70
60
50
40
30
20
10

80
60
40

(a) P1n (%)

20

Proton Number

Proton Number

β-DELAYED NEUTRON-EMISSION AND FISSION …

35

100
90
80
70
60
50
40
30
20
10

30
25
20
15

(a) P6n (%)

0
40

60

80 100 120 140 160 180 200 220

80
60
40

(b) P2n (%)

20
0

40

60

80 100 120 140 160 180 200 220

80
60
40

(c) P3n (%)

20

40

60

80 100 120 140 160 180 200 22040(d) P4n (%)

20
0

40

60

80 100 120 140 160 180 200 220

40(e) P5n (%)

20

40

60

15

(b) P7n (%)

10
5

40

60

80 100 120 140 160 180 200 220

35
30
25
20
15

(c) P8n (%)

10
5

40

60

100
90
80
70
60
50
40
30
20
10

80 100 120 140 160 180 200 220

35
30
25
20
15

(d) P9n (%)

10
5
0

0
20

20

20

80
60

25

100
90
80
70
60
50
40
30
20
10

100

Proton Number

Proton Number

20
100
90
80
70
60
50
40
30
20
10

35

0

80
60

80 100 120 140 160 180 200 220

30

20

100

Proton Number

Proton Number

20

60

0

0
100
90
80
70
60
50
40
30
20
10

40

100
90
80
70
60
50
40
30
20
10
20

100

Proton Number

Proton Number

20
100
90
80
70
60
50
40
30
20
10

5
0

20

100

Proton Number

Proton Number

20
100
90
80
70
60
50
40
30
20
10

10

40

60

100
90
80
70
60
50
40
30
20
10

80 100 120 140 160 180 200 220

35
30
25
20
15

(e) P10n (%)

10
5
0

80 100 120 140 160 180 200 220

20

Neutron Number

40

60

80 100 120 140 160 180 200 220

Neutron Number

FIG. 5. β-delayed neutron branching ratio of P1n to P5n in the
N-Z plane.

FIG. 6. Same as Fig. 5, but for P6n to P10n .

but show bends around neutron magic numbers N = 50, 82,
126, and 184. In general, nuclei with the magic numbers arestrongly bound as compared with neighboring nuclei, while
nuclei with a few more neutrons than the magic numbers are
relatively weakly bound. As a result, Sxn of nuclei with afew more neutrons than the magic numbers are low and it iseasy for those nuclei to emit neutrons from the highly excitedstates. Thus, Pxn tend to have a high value along the neutron
magic numbers, making the bends.
In the work of pn-RQRPA [15], only up to 5n emissions
are considered. However, through this study, we confirmedthat x > 5 should have been considered for nuclei close to
the neutron drip line. Figure 6 shows the Pxn (6  x  10).
Note that a different color scale from Fig. 5 is used to display
the results clearly. With increasing x, the distribution of Pxn
approaches the more-neutron-rich side, and the number ofnuclei with a prominent Pxn significantly decreases. In thecase of HFB-14 fission-barrier data, the number of nuclei
with Pxn > 10% is 356, 254, 197, 121, and 91 for x = 6,
7, 8, 9, and 10, respectively. For P10n only a limited nuclei
near the neutron drip line show a meaningful value, especiallyaround Z = 26 and N = 65, Z = 40 and N = 95, Z = 55 and044321-7

MINATO, MARKETIN, AND PAAR

PHYSICAL REVIEW C 104, 044321 (2021)

E strength function (MeV-1)

average difference, Mx (%)

3
2
1
0
-1

1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

1n

3n (a)

2n

1.0

140
Sb
139
Sb
138
Sb
137

(b)

0.8

1

2

3

4

5

6

7

8

9

Probability

-2
10

x

Sb

0.6
0.4

FIG. 7. Average difference Mx for various x defined in Eq. (16).
0.2

N = 135, and Z = 85 and N = 195, where the calculated Qβ
value reaches about 20 MeV.
To see the variations from the pn-RQRPA that uses thecutoff method, we compute the average difference defined as
N
1 
(HFM)
(cut)
Pxn
.
− Pxn
NN i

N

Mx =

(16)

(HFM)
(cut)
and Pxn
arewhere NN is the number of nuclei and Pxn
β-delayed branching ratios of the pn-RQRPA + HFM and thepn-RQRPA. The result is shown in Fig. 7. Upon increasing x,
Mx decreases monotonically and become negative for x  5.
This indicates that the BDNE branching ratios for low neutron
multiplicity are increased in the present approach, while they
are decreased for high neutron multiplicity.
To understand the difference of the results between the
pn-RQRPA + HFM and pn-RQRPA further, we plot the βstrength function of GT transition and xn emission energy
windows in Fig. 8(a) in case of 140 Sn as an example. We cansee that the most of the β strengths locate in the 1n emission
energy window. Since the amount of the β-strengths to xn
emission energy window directly becomes Pxn in the cutoff
method, P1n will be large for the pn-RQRPA.
On the other hand, in the pn-RQRPA + HFM, the BDNE
branching ratios are calculated through multiplying the βstrength functions by the isotope production ratios pi , as in
Eq. (10). Namely, the isotope production ratio mainly governs
the difference of Pxn between the pn-RQRPA + HFM and thepn-RQRPA. Figure 8(b) shows pi of Sb isotopes as a functionof the excitation energy of the daughter nucleus 140 Sb. Until
Sn = 2.3 MeV, pi of 140 Sb is unity. Namely, only γ deexcitation occurs for 140 Sb without emitting any neutron. Entering
the 1n emission energy window from E ∗ = Sn , pi of 139 Sb
suddenly becomes almost one and that of 140 Sb becomes almost zero. If the n-γ competition plays a role for this nucleus,
the curves of pi of 139,140 Sb at E ∗ = Sn become more smooth[55]. Upon further increasing the energy, pi of 139 Sb begins todecrease from S2n = 5.9 MeV and that of 138 Sb becomes dominant from E ∗ ≈ 7 MeV. A remarked point is that pi of 138 Sb
extend into not only the 2n emission energy window but the
3n emission energy window. This wide distribution is due to a

0.0

0

1

2

3

4

5

6

7

8

9

10

Excitation energy of 140Sb (MeV)

FIG. 8. (a) β strength function of 140 Sn. x-neutron (xn) emission energy windows are separated by the dotted lines. (b) Isotope
production ratios, pi of 137–140 Sb as a function of excitation energy,
assuming 1+ excited states of 140 Sb (the daughter nucleus of 140 Sn).
The calculation is performed by the HFM.

competition with 1n and 3n neutron-emission channels. However, the results of the pn-RQRPA and pn-RQRPA + HFM
will be almost the same because there are no significant βstrengths above the 2n emission energy window. Table II lists
Pxn of 140 Sn for the pn-RQRPA + HFM and the pn-RQRPA[15], in which the contributions from the FF transitions areincluded in addition to the GT transition. We can see that theresults of the pn-RQRPA + HFM and the pn-RQRPA show
a good agreement in this nucleus. Namely, the cutoff method
works well only if 0n and 1n emissions are the major decay
channel, like this case. We should mention that P0n and P1n
TABLE II. β-delayed neutron branching ratios (in units of %)of 140 Sn and 162 Sn calculated by the pn-RQRPA + HFM and thepn-RQRPA.
140

162

Sn

Sn

Model

pn-RQRPA+ HFM

pn-RQRPA

pn-RQRPA+ HFM

pn-RQRPA

P0n
P1n
P2n
P3n
P4n
P5n
P6n
P7n
P8n
P9n
P10n

62.6
36.8
0.6
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0

57.7
41.7
0.3
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0

12.5
34.2
12.9
17.7
7.1
13.6
1.2
0.7
0.1
0.0
0.0

10.4
19.1
15.7
20.2
1.5
9.4
1.4
20.1
0.6
0.8
0.8

044321-8

β-DELAYED NEUTRON-EMISSION AND FISSION …
2n

E strength function (MeV -1)

0.8
1n

0.7

4n
3n

5n

6n

7n

9n (a)

8n

0.6
0.5
0.4
0.3
0.2
0.1
1.0

(b)

0.8
Probability

PHYSICAL REVIEW C 104, 044321 (2021)

0.6

162
Sb
161
Sb
160
Sb
159
Sb
158
Sb
157
Sb
156

Sb

0.4
0.2
0.0

0

1

2

3

4

5

Excitation energy of

6

7

8

9

10

166

Sb (MeV)

FIG. 9. (a) β strength function of 162 Sn. xn emission energy
windows are separated by the dotted lines. (b) Isotope production
ratios of 162–156 Sb from 1+ excited states of 162 Sb (the daughter
nucleus of 162 Sn) as a function of excitation energy. The calculationis performed by the HFM.

are also largely different from those estimated by the cutoff
method if the n-γ competition would be significant [55].
As going to more neutron-rich nuclei, the difference between the pn-RQRPA and the pn-RQRPA + HFM becomesstriking. Figure 9 shows the β strength function of 162 Sn andthe isotope production ratios of Sb isotopes in the case of the
GT transition. The 1n-to-9n emission energy windows exist
closely up to E ∗ = 10 MeV and the 2n emission energy window is at much lower energy than 140 Sn. The most significant
β strengths are in the 7n emission energy window, so that P7n
is expected to be large in the cutoff method. On the other hand,looking at Fig. 9(b), pi of different isotopes distribute intoa wide range of excitation energy and are not stuffed insidea specific neutron-emission energy window. As mentioned,
this wide distribution ranging over various neutron-emission
energy windows is due to the competition with 1n, 2n, andmultineutron-emission channels. In the 7n emission energy
window, pi of 156–159 Sb isotopes exist side by side. Note thatpi of 155 Sb produced by 7n emission is about zero in the 7nemission energy window, so that P7n will be almost zero in the
pn-RQRPA + HFM. This mismatch comes about because thedaughter nucleus and descendant nuclei lost their excitationenergy every time neutrons are emitted. This never happensin the framework of the cutoff method. The results of Pxn
for the pn-RQRPA + HFM and the pn-RQRPA are listed inTable II, in which they show a large difference. We notice thatP7n = 20% for the pn-RQRPA, but 0.7% for the pn-RQRPA +HFM. The missing fraction of P7n is redistributed to Pxn withx  6. Note again that contributions from the FF transitions
are included in the result of Table II.

In the cutoff method, Pxn with a large x can have a finite
value if it is energetically allowed. However, excitation energyof nuclei is wasted every time neutrons are emitted. As aconsequence, the relative difference of Pxn between the cutoff
method and the HFM becomes larger as nuclei have lower Sxn
and higher Qβ that enable multineutron emission. As seen in
Fig. 7, Pxn for x  5 become negative due to this reason. The
reduced fractions in x  5 turn into Pxn for x  4. Since P10n
do not obtain a backward flow from higher x, the reduction islarger than others.
Figure 10 shows P1n of Pd (Z = 46) and Ag (Z = 47)
isotopes that are considered as important β-delayed neutron
precursors in the r process [16], and Os (Z = 76) and Ir (Z =77) isotopes. We plot the results of the FRDM + QRPA +HFM [58] for comparison. The result of pn-RQRPA + HFM
shows a P1n similar to that of the FRDM + QRPA + HFM fornuclei with small mass numbers. Upon going toward heavier
mass, the two calculations begin to differ. The noticeable
contrasts are found in nuclei close to the neutron drip line and210  A  230 for Os and Ir isotopes. A major factor affecting P1n is the β strength function and one neutron separationenergies Sn that are calculated from the theoretically predictedmass data, for which the pn-RQRPA + HFM and the FRDM+ QRPA + HFM use the global nuclear mass model [74]and FRDM2012 [84], respectively. For most of nuclei close
to the drip line, Sn of the global nuclear mass model is lower
than those of FRDM; for example, Sn of 240 Ir is 269 keV
for the global mass model and 690 keV for FRDM2012. To
check the sensitivity of P1n to mass data, we carry out thesame calculation replacing the mass data of the pn-RQRPA+ HFM with FRDM2012 and find that the result for nuclei
near the drip line globally becomes close to that of the FRDM+ QRPA + HFM. However, P1n in 210  A  230 for Os and
Ir isotopes still differ significantly. We thus consider that thedeviations found in 210  A  230 for Os and Ir isotopes aremainly attributed to the β strength function and those foundnear the neutron drip line is due to the mass data.
In this work, if daughter nuclei in the ground state areunstable against neutron emissions, i.e., for the case of Sn =0 MeV, we let them emit one neutron automatically. Therefore, P0n of corresponding nuclei become zero and the fraction
shifts to P1n . One can see the effect in Fig. 10, which shows
odd-even staggering for nuclei near the neutron drip line. For
example, P1n for 150 Pd is about 40%, while that for 149,151 Pd
is almost zero. The remarkably high fraction of P1n for 150 Pd
is because the daughter nucleus 150 Ag is a neutron-unboundnucleus according to the global nuclear mass model [74]. On
the other hand, 149,151 Ag is stable against neutron emission,
keeping P1n small.
We also computed delayed-neutron yield of thermalneutron-induced fission of 235 U. The result is shown in
Table III. The fission-fragment yields of thermal-neutroninduced fission of 235 U are taken from JENDL Fission
Product Yield 2011 (JENDL/FPY2011) [85,86]. The pnRQRPA gives the β-delayed neutron yield closest to theexperimental data among the four models because the Pn are
tuned to reproduce it. The result of pn-RQRPA + HFM also
reproduces the experimental data in the same order. Note thatthe calculated delayed neutron yields are an aggregated value

044321-9

(a)Pd(Z=46)

TABLE III. β-delayed neutron yield of thermal-neutron-induced
fission of 235 U. Fission-fragment yields are taken from JENDL/FPY2011 [85,86].

pn−RQRPA+HFM
pn−RQRPAFRDM+QRPA+HFM
exp.

P1n (%)

123 126 129 132 135 138 141 144 147 150 153 156
Mass A
110
100
90
80
70
60
50
40
30
20
10
0

(b)Ag(Z=47)

P1n (%)

120 123 126 129 132 135 138 141 144 147 150 153 156 159
Mass A
110
100
90
80
70
60
50
40
30
20
10
0

(c) Os(Z=76)

Mass A

P1n (%)

β-delayed neutron yield

pn-RQRPA
pn-RQRPA + HFM
GT2FRDM + QRPA + HFM
Expt. [87]1.43 × 10−2
1.00 × 10−20.81 × 10−20.81 × 10−2
(1.58 ± 0.05) × 10−2

underestimated and thus the calculated delayed neutron yieldis smaller than the experimental value.
The advantage of using the HFM is that one can estimate
the spectra of emitted particles. In particular, a high interest is
paid to delayed neutron spectra, which are expected to be used
for nondestructive analysis of nuclear materials [88,89]. However, it is difficult to measure delayed neutron spectra systematically for nuclei in the nuclear chart. Therefore, theoreticalpredictions are used to fill out the lack of the evaluated nuclear
data such as ENDF/B-VIII.0 [90] and forthcoming JENDL-5
if no experimental data are available. As an example, Fig. 11

1.6

200 205 210 215 220 225 230 235 240 245 250

110
100
90
80
70
60
50
40
30
20
10
0

Model

(d) Ir(Z=77)

DN Spectrum Dn (1/MeV)

110
100
90
80
70
60
50
40
30
20
10
0

PHYSICAL REVIEW C 104, 044321 (2021)

Rudstam+(1974)
pn-QRPA+HFM

1.2

0.8

0.4

(a)
0.0
0.0

0.5

3.2

200 205 210 215 220 225 230 235 240 245 250 255
Mass A

FIG. 10. β-delayed neutron branching ratio P1n of Pd, Ag, Os,
and Ir isotopes as a function of A. The experimental data are takenfrom Ref. [82].

summed over fission-fragment yields. Looking into important
precursor nuclei contributing the delayed neutron yield, theresult of pn-RQRPA is not necessarily correct. For example,
a precursor contributing the delayed neutron yield the most is137
I [P1n (e) = 7.66% [34]] according to the recent evaluation[82], while it is 91 Rb for the pn-RQRPA calculation and137
I enters the eighth place with P1n = 0.7%. On the other
hand, the pn-RQRPA + HFM calculation shows 137 I to be themost important precursor, however, the Pn (≈2.25%) is still

DN Spectrum Dn (1/MeV)

P1n (%)MINATO, MARKETIN, AND PAAR

1.0
1.5
2.0
neutron energy En (MeV)

89

Br

2.5

3.0

Shalev+(1974)
pn-QRPA+HFM

2.8
2.4
2.0
1.6
1.2
0.8

(b)

0.4
0.0
0.0

0.5

1.0
1.5
2.0
neutron energy En (MeV)

138

2.5

I
3.0

FIG. 11. BDNE spectrum of (a) 89 Br and (b) 138 I. The calculated
result is shown by the solid line (red). Experimental data are takenfrom Rudstam et al. [91] and Shalev et al. [92].

044321-10β-DELAYED NEUTRON-EMISSION AND FISSION …

PHYSICAL REVIEW C 104, 044321 (2021)

110
0

20

40

60

80

100

0

20

40

60

80

100

Z

100

(a) HFB-14(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(a) HFB-14(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90

230

240

(b) FRDM(f)
230

240

110
Z

100

(c) ETFSI(n)

90

230

240

(c) ETFSI(f)
230

240

110
Z

100

(d) SBM(n)

90

230

240

(d) SBM(f)

N

230

240

N

FIG. 12. BDNE branching ratios Pn (left panels) and BDF branching ratios Pf (right panels) in unit of % calculated by the fission barrier
data of (a) HFB-14 [76], (b) FRDM [78], (c) ETFSI [77], and (d) SBM [79]. Nuclei where no fission barrier or β-decay data are provided are
given as blank.

shows the BDNE spectrum of 89 Br and 138 I, which are typical
β-delayed neutron precursors. We also plot the experimental
data taken from Refs. [91,92]. Although fine structures observed in the experimental data are not reproduced well by the
pn-RQRPA + HFM, the computed results reasonably emulate
the experimental β-delayed neutron spectra.
B. Effect of β-delayed fission

Figures 12(a)–(d) show BDNE and BDF branching ratios
for Z  90 in the N-Z plane using different fission barrier
data of HFB-14 [76], FRDM [78], ETFSI [77], and SBM,
respectively [79]. Note that nuclei where no fission barrier or
β-decay data are provided are given as blank. As is obvious,
for nuclei where Pf is significant, Pn is small since they arecorrelated. We can see that Pn and Pf greatly vary among thefission barrier data. A common feature is that BDF branchingratios become high around 93  Z  110 and 184  N 
200 for FRDM, ETFSI, and SBM. Although the number ofnuclei calculated in this region is limited for HFB-14, Pf
become meaningful as well around Z = 100 and N = 190.
BDF is also significant from Z = 91, N = 160 to Z = 100,
N = 185 for FRDM, ETFSI, and SBM, while that for HFB14 is negligibly small. There exists an odd-even dependence
for Pf in the region of N = 160 to 185, in particular, for
FRDM and SBM. This odd-even dependence is also found in
Ref. [26], which calculates Pf with the FRDM + QRPA andthe FRDM barrier data [78]. An effect of the neutron magic
number of N ≈ 184 is seen for ETFSI and weakly for FRDM
and SBM. On the neutron-rich side, Pn become major for
HFB-14, ETFSI, and SBM, while BDNE and BDF competewith each other for FRDM.
Figure 13 shows P1n and P1n f calculated with the different
fission barrier data for Z  90 in the N-Z plane. The numberof nuclei showing a significant P1n f is much less than Pf .
This means that most nuclei do fission directly after β decay
without emitting neutrons in a high probability. However,
FRDM still has relatively many nuclei with prominent P1n f
as compared with the other fission barrier data. As will be
discussed later, this is because fission barrier of FRDM is low
and single-humped. Figure 14 shows the result of P2n and P2n f .
The number of nuclei having a significant P2n f is even lessthan that of P1n f . In particular, there are few nuclei showing aprominent P2n f for HFB-14 and SBM, and accordingly theirdistributions of P2n become similar to each other. On the other
hand, there are still a lot of nuclei with prominent P2n f in theneutron-rich region for FRDM. ETFSI also shows some nucleithat have compelling P2n f .
Figure 15 shows the result for P3n f . For HFB-14 and SBM,
only a few nuclei show a prominent P3n f . Thus the distributionof P3n is almost the same. In our calculation, P3n f of all nuclei
are less than 10% for HFB-14, ETFSI, and SBM, while there
are still a number of prominent P3n f in neutron-rich sides for
FRDM. We also plot P4n and P4n f in Fig. 16. There are acouple of nuclei that give a significant P4n f for ETFSI andno compelling P4n f for HFB-14 and SBM. Even for FRDM,
only limited nuclei show a prominent P4n f . In Fig. 17, no
significant P5n f is observed for HFB-14, ETFSI, and SBM,
and the results of P5n are almost the same at least for the nuclei
where the calculation is done. For FRDM, there are still nucleithat have a prominent P5n f , however, the number is decreased
from P4n f . We confirm that Pxn f (x  6) are less than 1% for
HFB-14, ETFSI, and SBM, while those for FRDM are stilllarge, especially for nuclei with odd Z from 93 to 103. An
importance of multichance fission has been pointed out using
the FRDM fission barrier data [26]. However, it is seen fromthis work that the importance of multichance fission strongly
depends on fission barrier data.

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PHYSICAL REVIEW C 104, 044321 (2021)

110
0

20

40

60

80

100

0

20

40

60

80

100

Z

100

(a) HFB-14(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(a) HFB-14(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90

230

240

(b) FRDM(f)
230

240

110
Z

100

(c) ETFSI(n)

90
160

170

180

190

200

210

220

230

240

(c) ETFSI(f)
160

170

180

190

200

210

220

230

240

110
Z

100

(d) SBM(n)

90
160

170

180

190

200

210

220

230

240

(d) SBM(f)
160

170

180

190

N

200

210

220

230

240

N

FIG. 13. Same as Fig. 12, but for P1n f .

data around Z = 100, N = 170, while they are strongly hindered for the neutron-rich region. HFB-14 has triple fission
barriers. Although the inner barrier height decreases fromN = 160 to 178, the middle and outer barriers emerge fromN = 170. This triple-humped structure strongly hinders thefission rates and Pf in this region, as we have seen in Fig. 12.
Around N = 190, the inner and middle barrier heights are low
and the outer barrier is irrelevant so that Pf become significant
in this region. With increasing neutron number, the inner and

To understand the result of Pxn f more qualitatively, we plot
fission barrier heights (B f ) of fermium isotopes used in thiswork in Fig. 18. The numbers in parentheses for the HFB-14
indicate the (1) inner, (2) middle, and (3) outer fission barriers.
Similarly, the numbers in parentheses for the ETFSI indicate(1) inner and (2) outer fission barriers. The fission barrier of
SBM is as high as the other fission barrier data for 160  N 
185, while it becomes significantly larger above N = 186.
Thus, Pf of SBM are comparable to the other fission barrier
110
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40

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90
160

170

200

210

220

230

240

(a) HFB-14(f)
160

170

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90
160

170

180

190

200

210

220

230

240

(b) FRDM(f)
160

170

180

190

200

210

220

230

240

110
Z

100

(c) ETFSI(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(c) ETFSI(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(d) SBM(n)

90

230

240

(d) SBM(f)N

N

FIG. 14. Same as Fig. 12, but for P2n f .
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20

40

60

80

100

0

20

40

60

80

100

Z

100

(a) HFB-14(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(a) HFB-14(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90

230

240

(b) FRDM(f)
230

240

110
Z

100

(c) ETFSI(n)

90
160

170

180

190

200

210

220

230

240

(c) ETFSI(f)
160

170

180

190

200

210

220

230

240

110
Z

100

(d) SBM(n)

90
160

170

180

190

200

210

220

230

240

(d) SBM(f)
160

170

180

190

N

200

210

220

230

240

N

FIG. 15. Same as Fig. 12, but for P3n f .

N = 183 would also have a meaningful effect on forming thevalley. Above N = 192, the inner barrier of the ETFSI showsa behavior similar to that of HFB-14. However, Pf of ETFSI islarger than that of HFB-14 because ETFSI is a single-humped
barrier while HFB-14 gives a double- or triple-humped fission
barrier in this region. The barrier heights of FRDM are rather
constant (B f ≈ 4 MeV) except in 184  N  200, where B f
are close to the inner and middle barriers of HFB-14. In spiteof that, the reason why the FRDM has more prominent Pxn f

middle barriers become high again and the fission rates areagain hindered. Similar to HFB-14, the inner barrier of ETFSI
decreases from N = 160 to 172. However, the barrier heights
are about 2 MeV lower than HFB-14 and BDF competes
with BDNE. As a result, Pf for ETFSI are significant in thisregion, in contrast with HFB-14. From N = 173, the innerbarrier increases and peaks around N = 183. This fact makes
Pf of ETFSI less important, showing the valley in this region
as seen in Fig. 12. The outer barrier which emerges from
110
0

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40

180

190

60

80

100

0

20

40

180

190

60

80

100

Z

100

(a) HFB-14(n)

90
160

170

200

210

220

230

240

(a) HFB-14(f)
160

170

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90
160

170

180

190

200

210

220

230

240

(b) FRDM(f)
160

170

180

190

200

210

220

230

240

110
Z

100

(c) ETFSI(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(c) ETFSI(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(d) SBM(n)

90

230

240

(d) SBM(f)N

N

FIG. 16. Same as Fig. 12, but for P4n f .
044321-13

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60

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0

20

40

60

80

100

Z

100

(a) HFB-14(n)

90
160

170

180

190

200

210

220

230

240

(a) HFB-14(f)
160

170

180

190

200

210

220

230

240

110
Z

100

(b) FRDM(n)

90
160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

(b) FRDM(f)
160

170

180

190

200

210

220

160

170

180

190

200

210

220

160

170

180

190

200

210

220

230

240

110
Z

100

(c) ETFSI(n)

90

230

240

(c) ETFSI(f)
230

240

110
Z

100

(d) SBM(n)

90

230

240

(d) SBM(f)

N

230

240

N

FIG. 17. Same as Fig. 12, but for P5n f .

in this region is that HFB-14 as well as ETFSI have a doubleor triple-humped barrier. The importance of BDF is thus notdetermined only by the fission barrier height but by the number of barrier humps. The SBM also provides a single barrier,however, the barrier height is larger than for the FRDM and
Pf are significantly reduced.
C. β-delayed α emission

In this work, we allow daughter nuclei also to decay
by α-particle emission and study the β-delayed α emission
branching ratio Pα as well as BDNE and BDF. The calculated
Pα are shown in Fig. 19. Note that the maximum scale is set
to be Pα = 10% for illustration. Because we found that nuclei
where β-delayed α emission is prominent are not affected by

12
10
8

HFB-14(1)
HFB-14(2)HFB-14(3)FRDM
ETFSI(1)ETFSI(2)SBM

Fm(Z=100)

6
4
2
0
160

170

10

110
Proton Number

Fission Barrier Height (MeV)14

the selection of fission barrier data, the results in the caseof the FRDM fission barrier data are plotted. Since neutron
emission overcomes α-particle emission in the neutron-richregion, a non-negligible Pα is observed only in a band ofnear the valley of stability. To our knowledge, experimental
data on β-delayed α-particle emission has been reported only
for 214 Bi (Pα = 0.003%) [81,93] in the range of Fig. 19. The
calculated result of Pα for 214 Bi is 0.026%. Although ourmodel overestimates the experimental data, it shows that the
β-delayed α-particle emission branching ratio is very small.
Exceptional cases exceeding Pα = 10% are 210,211 Bi (Z =83) and 248 Am (Z = 95), that have Pα = 92.5%, 48.9%,
and 15.9%, respectively. The daughter nuclei 210,211 Po and248
Cm have a relatively high α-decay rates from the groundstates [93]. It is not thus surprising that they do α-particle
emission following β decay, competing with γ and otherdeexcitations. We checked delayed proton emission branching

180

190
200
Neutron Number

210

220

6
90
4
80

2

70

230

FIG. 18. Fission barrier height of Fm (Z = 100) isotopes for
HFB-14 [76], FRDM [78], ETFSI [77], and SBM [79] models. The
numbers in parentheses indicate 1 : inner, 2 : middle, and 3 : outerfission barrier heights.

8

100

0
120

130

140

150
160
170
Neutron Number

180

190

FIG. 19. Delayed alpha branching ratios Pα (%) in the case of
FRDM fission barrier data. The black filled squares indicate stableor long-lived nuclei.

044321-14

<n>

<n>

9
8
7
6
5
4
3
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8
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0

<n>

100
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60
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20
10

PHYSICAL REVIEW C 104, 044321 (2021)

12
9
6
3
0

<n>

Proton Number

β-DELAYED NEUTRON-EMISSION AND FISSION …

12
9
6
3
0

20 40 60 80 100 120 140 160 180 200 220
Neutron Number

D. Number and energy of emitted neutrons

From the β-delayed neutron branching ratios and spectra,
we can also calculate the mean number and the mean kinetic
energy of emitted neutrons that are defined as(17)x(Pxn + Pxn f + Pxnα + Pxnp )n =xEn Dn (En )dEn ,

35

(18)

respectively. Those quantities are important for examining the
r-process final abundance after the freeze-out phase. Note thatneutrons can be also provided from highly excited fission
fragments, known as prompt neutrons. However, the results
discussed in this work are limited to β-delayed neutrons since
the prompt neutrons are beyond our scope.
Figure 20 shows the mean number n of β-delayed neutrons in the N-Z plane. As expected, the value of n near
the valley of the stability is small, while it becomes prominent as the neutron number increases. In our calculation, themaximum n is about 9.7 for 192 La, which is in the spotshowing a prominent Pxn even for a large x, as we haveseen in Fig. 6. We also plot the mean neutron numbers fornickel, tin, and fermium isotopes for the pn-RQRPA + HFM
and the pn-RQRPA in Fig. 21. For most nuclei, the mean
neutron number is less than 1. From the appreciably-neutronrich side where the neutron number is approximately double
the proton number, n begins to increase significantly, being
more than 1. The difference between the pn-RQRPA + HFM
and the pn-RQRPA is not large for nuclei with a relatively
small A, while n of the pn-RQRPA + HFM becomes significantly smaller than those of pn-RQRPA as the neutron
number increases, because excitation energies to be used forneutron emissions are reduced effectively by β-delayed neutrons, as discussed in the previous section. For nuclei at theneutron drip line, the deviations of n from the pn-RQRPA

40
Sn(Z=50)

45

50

75
80
85
Fm(Z=100) 0.2

90

55

95

60

70

100 105 110 115 120
FRDM
ETFSI
SBM

0.1
0
185

65

190

195

200

170 175 180 185 190 195 200 205 210 215 220 225 230
Neutron Number

FIG. 21. Mean number of emitted neutrons for nickel, tin, andfermium isotopes.

that uses the cutoff method are approximately a factor oftwo.
We also plot the results with different fission barrier data
for fermium isotopes (185  N  200) in the inset of Fig. 21.
The mean neutron number of SBM is the largest in this mass
region, although the variation of n among the fission barrier
data is rather small. This is because the fission rates of SBM
are low due to the high fission barriers as seen in Fig. 18,
and the chances for emitting more neutrons are enhanced
instead of fission. The difference among fission barrier data
becomes negligible as going to the neutron drip line because
Pf are commonly much smaller than Pn . Figure 22 shows themean energy of β-delayed neutrons. In the region of lightnuclei around 8  Z  28, the mean energy is relatively large,
while it becomes smaller as going to heavier nuclei. The
average energy En for Z  28 is 935 keV, while that forZ > 28 is 432 keV. In addition, En tend to be high for nuclei
just above the neutron magic number, especially for N = 28,
50, and 82.
The kinetic-energy distribution of an emitted neutron,
Xn ( n ), is approximately expressed byXn ( n ) ∝ ρN (E ∗ − n − Sn )ρn ( n ),(19)

where ρN (E ) and ρn ( n ) are the level density of decayed
nuclei with an excitation energy of E and the phase space ofthe emitted neutron with the kinetic energy n , respectively. In

Proton Number

ratios Pp as well; however, no significant Pp is obtained inthis work. Note that our calculation is carried out only for
nuclei in the β-stability line to a neutron-rich region. Our
future plan is to study a neutron-deficient region where itis expected that Pα as well as Pp become more important
than Pn .

En =pn-RQRPA
pn-RQRPA+HFM

70

FIG. 20. Mean number of emitted neutrons in the N-Z plane. The
calculation is performed by the HFB-14 fission barrier data [76].

andNi(Z=28)

3.0

100
90
80
70
60
50
40
30
20
10

044321-15

2.5
2.0
1.5
1.0

<En>

0.5
0.0

20

40

60

80 100 120 140 160 180 200 220
Neutron Number

FIG. 22. Mean energy of emitted neutrons in the N-Z plane.

<En>

<En>

MINATO, MARKETIN, AND PAAR
2
Ni(Z=28)
1.5
1
0.5
0
35
40
1.5
Sn(Z=50)
1

50

85

90

55

60

65

105

110

70

0.5
0
70

<En>

45

PHYSICAL REVIEW C 104, 044321 (2021)

75

80

95

100

115

120

0.4 Fm(Z=100)
0.3
HFB-14
FRDM
0.2
ETFSI
0.1
SBM
0
170 175 180 185 190 195 200 205 210 215 220 225 230
Neutron Number

FIG. 23. Mean energy of β-delayed neutron En . For fermium
isotopes, the results with different fission barrier data are also plotted
together.

general, the level density exponentially grows as the excitationenergy increases. Thus a transition to higher levels is themost likely, however, the phase space of the outgoing neutron
becomes small instead. For light nuclei and nuclei around theneutron magic numbers, the level densities are relatively low
even at a high excitation energy and thus emitted neutrons canhave a large phase space easily. On the other hand, the level
densities of heavy nuclei rapidly increase as the excitationenergy goes up, the phase space of neutron is limited, and the
kinetic energy becomes low.
We plot the mean energy for nickel, tin, and fermium
isotopes in Fig. 23. For comparison, we also plot the resultswith different fission barrier data in the case of fermium
isotopes. For nickel isotopes, En gradually increase andbecome relatively high for N = 50, 52, and 54. The highmean energies are mainly due to the N = 50 magic number,as discussed above. Upon increasing the neutron number,
En almost become constant around 1 MeV. For tin isotopes,
we can see again that the mean energy is locally high for
N = 84 just above the neutron magic number N = 82. The
mean energy becomes En ≈ 0.6 MeV upon increasing theneutron number. For fermium isotopes, En are about 0.3
to 0.4 MeV, which are only about one-third of nickel isotopes and half of tin isotopes. We can also see the fission
barrier dependence of En around 185 < N < 200. However, the variations are not large because the mean energies
of delayed neutrons do not vary significantly with neutron
number.
IV. SUMMARY

In this work, we calculated the BDNE and BDF bycombining the discrete β strength function provided by the
pn-RQRPA [15] and Hauser-Feshbach statistical model [56].
We could improve P1n over a wide range of nuclei as comparedwith the preceding work of Ref. [15] and obtained the root
mean square of P1n , which is comparable to other precedingstudies upon adjusting a phenomenological width parame-

ter. The different role of Gaussian- and the Lorentzian-type
weight functions was also discussed.
We next discussed partial β-delayed neutron branching ratios Pxn . With increasing x, the distribution of Pxn approaches
the more-neutron-rich side, and the number of nuclei witha prominent Pxn significantly decreases. We also studied thevariations from the pn-RQRPA calculations that use the cutoff
method. We concluded that the cutoff method works reasonably well only if 0n and 1n emissions are the main decay
channels, while isotope production ratios calculated by the
HFM calculation becomes important to obtain an exact result for neutron-rich nuclei that emits multiple β-delayed
neutrons. Calculated β-delayed neutron spectra of 89 Br and138
I were compared with the experimental data and it turned
out that the pn-RQRPA + HFM was able to emulate theexperimental data although some fine structures were not reproduced well. We also calculated the β-delayed neutron yieldof thermal-neutron-induced fission of 235 U. The computed
result underestimated the experimental data by about 35%.
We pointed out that the discrepancy was due to small Pn ofimportant precursors. An improvement is expected if the Pn of
those nuclei are better reproduced.
BDF branching ratios were calculated with four fission barrier data. By comparing the results of the different barrier data,
BDF commonly became important around 93  Z  110 and
184  N  200. It was also found that most nuclei fission
directly with a high probability after β decay without emitting
neutrons. However, we confirmed that both Pf and Pxn f for
most nuclei vary greatly over the fission barrier data used.
Taking fermium isotopes as an example, we discussed therelation between the BDF branching ratios and barrier heights,
and it was found that the BDF are sensitive not only to thebarrier height but also to the number of barrier humps. The
FRDM results in a single-humped barrier of relatively low
height, so that Pf and Pxn f are larger than the other fission
data, while HFB-14 and ETFSI give a multihumped barrier,so that Pf and Pxn f are smaller than the other fission data.
However, the number of nuclei given in HFB-14 and ETFSI
is smaller than FRDM and SBM. For this reason, it is difficult to choose the best BDF data that can be recommended
from our results. However, the products with four barriers
allow further sensitive studies of r-process nucleosynthesison the nuclear fission. To accurately assess the role of BDF
in the r process, detailed information (fission path, curvature, the number of barrier hump, etc.) and an extension to awide range of nuclei based on microscopic models are highly
required.
We also discussed the mean number and energy of delayed neutrons. For most nuclei, the mean number of delayed
neutrons is less than 1. As the ratio of neutron number toproton number becomes appreciably large, the mean numberof delayed neutrons increases. We also studied the fission
barrier dependence and found that the mean number becomeshigh if BDF is suppressed. The mean number for pn-RQRPA+ HFM was smaller than that of pn-RQRPA because excitation energies were reduced by β-delayed neutrons. The
mean energy of delayed neutrons was large for light nuclei
and nuclei close to the magic numbers. It was explained that

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β-DELAYED NEUTRON-EMISSION AND FISSION …

PHYSICAL REVIEW C 104, 044321 (2021)

this was governed by the interplay of the level densities of
decayed nuclei and the phase space of emitted neutrons. Upongoing to neutron-rich nuclei, the mean energy becomes almost
constant.
We should comment on several subjects for future work.
One of them is level structures of neutron-rich nuclei whichwere calculated by the phenomenological method as described in Sec. II A because they are not known. Moreaccurate data of BDNE and BDF are expected when the levelstructures are investigated in the future. We also calculated
odd-mass nuclei in the same way as even-mass nuclei, namely,
we just imposed the expectation value of the particle number
operator to be odd in the RHB calculation. Applying theequal-filling approximation and the blocking approximation,
which are more advanced ways to treat odd mass nuclei, may
improve the present result. Taking into account continuum
states, which become important especially for neutron-richnuclei, as well as the shape deformation, will also improve
the accuracy of BDNE and BDF calculations. Our discussion
in this work is limited to neutron-rich nuclei. However, it will
be interesting to extend our framework to neutron-deficient
nuclei where it is expected that Pα as well as Pp become more
important than neutron-rich nuclei.
A table of BDNE, BDF, and β-delayed α-particle emission
branching ratios calculated in this work is available in the
Supplemental Material [62].

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ACKNOWLEDGMENTS

N.P. acknowledges support by the QuantiXLie Centre of
Excellence, a project co financed by the Croatian Government and European Union through the European Regional
Development Fund and the Competitiveness and Cohesion
Operational Programme (KK.01.1.1.01).

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